As can be seen from SaintRidley's list of early Germanic language derivatives, "penny" is a word that has been in use for even longer than the English language has existed; as such, it's original derivation has been lost. It seems likely that it is, at least in part, an onomatopoeia - a word derived from the sound an object makes. If you take a pile of mediaeval silver pennies - these coins are rather small and very thin - and pour them out on a table, they make a kind of "ping-ping-ping" sound. The "ning" ending is diminutive, thus "penning" would mean "small object that makes a tinny, ringing sound".
Equally obscure, to me at least, is the usage of the word "penny" in North America as a slang term for the one cent coin. The Americans almost certainly adopted it first, since "real pennies", the large copper coins from Britain or the local penny tokens the same size, were still in circulation in Canada up until the 1850s. It's particularly confusing given that the original large cents, in both Canada and America, are closer in size and equivalent value to the halfpenny rather than the penny.
The word "dime", on the other hand, is a purely North American invention. Decimal currency, as originally envisaged by the American revolutionaries, was exactly what its name implies: base-10 counting. Thus, 10 mils to the cent, 10 cents to the dime, 10 dimes to the dollar, 10 dollars to the eagle. This replaced the somewhat random allocation of numbers in the Sterling currency system
inherited from the British, of 4 farthings to a penny, 12 pence to the shilling, 5 shillings to the crown and 4 crowns to the pound.
Since 10 is easily divisible by 5 and 2, the most logical intermediate denominations for coins in a decimal system are 2 and 5. Thus, 1 cent, 2 cents, 5 cents, 1 dime, 2 dimes, 5 dimes, 1 dollar, 2 dollars, 5 dollars, and so forth. However, the "dollar", as adapted directly from the Spanish dollar, was already circulating in America with fractional coinage based on eighths, rather than tenths. Thus, the Americans found it easier to issue a "quarter-dollar" coin on par with the Spanish 2 reales coins already circulating, rather than a 2 dimes (20 cents) coin.
Don't say "infinitely" when you mean "very"; otherwise, you'll have no word left when you want to talk about something really infinite. - C. S. Lewis