In science, the gas constant, symbol "R", is the proportionality constant in the ideal gas law having a value of: $~ 8.314 \frac{J}{K \cdot mol} ~$

This modern value for the gas constant was assigned in YEAR (add).

History
The first dominant use of R as a constant in the various gas laws seems to have been the use by French engineer Emile Clapyron's use in his 1834 Memoir on the Motive Power of Fire, where he states that Mariotte’s law (PV = k, at constant temperature) combined with that of Gay-Lussac's law (P = kT, at constant volume), the latter of which being derived by French chemist Joseph Gay-Lussac in 1802, gives the expression: $~ Pv = R(267 + t) ~$

This is sometimes referred to as the 'G-M law' or the 'law of Marriatte and Gay-Lussac'.

In 1850, German physicist Rudolf Clausius updated the work of Clapeyron using the 1847 work of French chemist Henri Regnault, where he reevaluated the content inside of the parentheses, rewriting the gas equation as: $~ Pv = R(273 + t) ~$

In 1864, Clausius further simplified this expression by using the absolute temperature scale, conceived by Irish physicist William Thomson in 1848: $~ T = 273 + t ~$

where t is in degrees centigrade and T is in Kelvin, such that with substitution in the second previous equation, by 1864 he had arrived at: $~ pv = RT ~$

The first person to convert R into a universal gas constant, according to American chemistry historian William Jensen, was German chemist August Horstmann who in 1873 rewrote the previous equation as: $~ up = RT ~$

where u is the "volume of molecular weight [mole] of the gas" and where R is "the constant for the G-M law with regard to the molecular volume [molar volume]."  In this sense, if u is taken as volume per number of particles n (on the original 1738 use by Daniel Bernoulli), then we would have: $~ \frac{Vp}{n} = RT ~$
or $~ PV = nRT ~$

This later version of the equation, with the explicit use of "n" as the number of mols, seems to have been first use in the classic 1923 textbook Thermodynamics and the Free Energy of Chemical Substances by American physical chemists Gilbert Lewis and Merle Randall. 

Symbol etymology
See main: R (etymology section)
The origin of the symbol R, prior to its 1834 use by Clapeyron, is difficult to pin down. This seems to be due to the fact often early versions of the gas laws were simply stated verbally.

References
1. Daintith, John. (2005). Oxford Dictionary of Physics. Oxford University Press.
2. Clapeyron, Émile. (1834). “Memoir on the Motive Power of Heat”, Journal de l’Ecole Polytechnique. XIV, 153 (and Poggendorff's Annalender Physick, LIX,  446, 566).
3. Clausius, Rudolf. (1850). “On the Moving Force of Heat and the Laws of Heat which may be Deduced Therefrom” (pg. 21), Communicated to the Academy of Berlin, Feb.; Published in Poggendorff’s Annalen, March-April, Vol. lxxix, pgs. 368, 500, and Translated in the Philosophical Magazine, July 1851, Vol. ii. pgs. 1, 102.
4. Jensen, William B. (2003). “The Universal Gas Constant R” (abstract: “this column traces the history of the gas constant R and the probable reason for its representation by the letter R.), J. Chem. Edu. 80: 731.
5. Horstmann, August F. (1973). “Theorie der Dissociation”, Liebig’s Annalen der Chemie und Pharmacie, Bd. 170 (CLXX), 192-210.
6. Lewis, Gilbert N. and Randall, Merle. (1923). Thermodynamics and the Free Energy of Chemical Substances, McGraw-Hill Book Co., Inc.