Here is a Greek word for Prayer - προσεύχομαι

G4336 (Mickelson's Enhanced Strong's Greek and Hebrew Dictionaries)
G4336 προσεύχομαι proseuchomai (pros-yoo'-khom-ai) v.
1. to pray to God, i.e. supplicate, worship
[from G4314 and G2172]
KJV: pray (X earnestly, for), make prayer 
Root(s): G4314, G2172 
Compare: G2172, G4727 

I'm reluctant to take the first defintion because of our rule about "Biblical" Greek". In addition to that, the first definition says, " to pray to God, i.e. supplicate, worship", but combination of the root words (below) doesn't bear out that definition. So, to get a better idea of what this word means, let's look at the Root words: G431 (πρός)4, G2172 (εὔχομαι)

Here is the first root word - πρός

• G4314 πρός pros (pros') prep.forward to, i.e. toward
• (genitive case) the side of, i.e. pertaining to
• (dative case) by the side of, i.e. near to
• (accusative case, usually) the place, time, occasion, or respect (which is the destination of the relation, i.e. whither or for which it is predicated)

[a strengthened form of G4253, a preposition of direction]
KJV: about, according to , against, among, at, because of, before, between, (where-)by, for, X at thy house, in, for intent, nigh unto, of, which pertain to, that, to (the end that), X together, to (you) -ward, unto, with(-in). In the comparative case, it denotes essentially the same applications, namely, motion towards, accession to, or nearness at 
Root(s): G4253 

Notice how the top three definitions talk about "forward to, i.e., toward" and " the side of, i.e. pertaining to" and " by the side of, i.e. near to" Add to this the avove phrase: "[a strengthened form of G4253, a preposition of direction]"

Here is the second root word - εὔχομαι

G2172 εὔχομαι euchomai (yoo'-khom-ai) v.

1. 1. to wish
2. 2. (by implication) to pray to God

[middle voice of a primary verb]
KJV: pray, will, wish 
Compare: G4336, G4727 
See also: G685 

Notice how the top definition says "to wish". We don't use the second definition because of our thoughts about "Biblical" Greek and because it is said to be an implication, not a real definiton.

Putting it All Together

Now when we look at the Greek word for prayer "προσεύχομαι", which is a combination of "G4314 πρός" and "G2172 εὔχομαι" we see a clarified meaning of the word for prayer. "pertaining to/toward (a preposition of direction)" + "to wish". So, when we pray, we set our minds "toward/pertaining to + to wish".

Here is Another Greek word for prayer: παρακαλέω

G3870 (Mickelson's Enhanced Strong's Greek and Hebrew Dictionaries)
G3870 παρακαλέω parakaleo (par-ak-al-eh'-o) v.
1. (properly) to call near (as to receive or give help)
2. to implore, entreat (to urgently ask (for mercy or help))
3. to exhort (to urgently counsel, encourage or admonish)
4. to comfort
[from G3844 and G2564]
KJV: beseech, call for, (be of good) comfort, desire, (give) exhort(-ation), intreat, pray
Root(s): G3844, G2564

Here one of the root words: παρά

G3844 παρά para (par-ah') prep.

1. (properly) near

2. (genitive case) from beside (literally or figuratively)

3. (dative case) at (or in) the vicinity of (objectively or subjectively)

4. (accusative case) to the proximity with (local (especially beyond or opposed to) or causal (on account of))

[a primary preposition]

KJV: above, against, among, at, before, by, contrary to, X friend, from, + give (such things as they), + that (she) had, X his, in, more than, nigh unto, (out) of, past, save, side...by, in the sight of, than, (there-)fore, with. In compounds it retains the same variety of application

Compare: G3326

Here is the second root word: καλέω

G2564 καλέω kaleo (kal-eh'-o) v.
1. to "call" (aloud)
{(properly) aloud, but used in a variety of applications, directly or otherwise}
[akin to the base of G2753]
KJV: bid, call (forth), (whose, whose sur-)name (was (called))
See also: G2753

Notice how the definition very clearly says "to call (aloud)".